A new class of weakly K -analytic Banach spaces

A new class of weakly K -analytic Banach spaces

S. Mercourakis, E. Stamati

Abstract. In this paper we define and investigate a new subclass of those Banach spaces which areK-analytic in their weak topology; we call them strongly weaklyK-analytic (SWKA) Banach spaces. The class of SWKA Banach spaces extends the known class of strongly weakly compactly generated (SWCG) Banach spaces (and their subspaces) and it is related to that in the same way as the familiar classes of weaklyK-analytic (WKA) and weakly compactly generated (WCG) Banach spaces are related.

We show that: (i) not every separable Banach space is SWKA; (ii) every separable SWKA Banach space not containing ℓ¹ is Polish; (iii) we answer in the negative a question posed in [S-W] by constructing a subspace X of the SWCG space L¹[0,1]

which is not SWCG.

Keywords: WKA, SWKA Banach spaces,K-analytic space, Baire space, Polish space Classification: Primary 46B20, 54H05, 03E75; Secondary 46B26

Introduction

The purpose of the present paper is to introduce and study a new class of Banach spaces which areK-analytic in their weak topology ([T], [N]), that extends the class of strongly weakly compactly generated (SWCG) Banach spaces (and their subspaces) introduced by Schl¨uchtermann and Wheeler in [S-W]. We call them strongly weaklyK-analytic-SWKA Banach spaces (see Definition 1.4).

Our results show that the classes of SWCG and SWKA Banach spaces are related in the same way as the known classes of weakly compactly generated (WCG) and weaklyK-analytic (WKA) Banach spaces.

In Section 1, we study the general properties of SWKA Banach spaces. We show in particular, that every subspace of a SWCG Banach space is SWKA (Proposition 1.5) and also, that a Banach space with separable dual is SWKA iff it is Polish (Proposition 1.9). So we get as an easy consequence that not every separable Banach space is SWKA (Corollary 1.10). We also give a useful charac- terization of SWKA Banach spaces by using the pointwise order of the Baire space Σ =N^N(Proposition 1.7), which is similar to a corresponding characterization of WKA Banach spaces due to Talagrand ([T, Proposition 6.13]).

In the same section we strengthen the classicalK-analytic property for topolog- ical spaces by introducing the notion of a stronglyK-analytic topological space (Definition 1.11), so that a Banach space X is SWKA iff X is a strongly K- analytic topological space in its weak topology. Then we characterize strongly

K-analytic topological spaces as those topological spacesX for which the space K(X) of non empty compact subspaces ofX endowed with Vietoris topology is a (strongly)K-analytic topological space (Theorem 1.12).

In Section 2, we show, by using results of Stegall (Theorem 2.3) and Stern (The- orem 2.2), that every separable SWKA Banach space not containingℓ¹ is Polish (Theorem 2.6). This result generalizes a result from [S-W, Proposition 2.10]: Ev- ery (subspace of a) SWCG space not containingℓ¹ is reflexive. We also give an example of a separable dual space X₀ with an unconditional basis (hence X₀ is weakly sequentially complete) which is not SWKA (Example 2.9); we mention, in contrast, that every (subspace of a) SWCG space is weakly sequentially complete ([S-W, Theorem 2.5]).

The third section of the paper is devoted to the construction of a closed lin- ear subspace X of the SWCG space L¹[0,1] (thus X is SWKA) which is not a SWCG space. This construction is closely related to the classical construction of Rosenthal [R1] of a non WCG subspace of the spaceL¹[0,1]^c(c= the cardinality of the continuum). The present example of the spaceX answers a question posed in [S-W].

We would like to thank S. Argyros for suggesting us a result of J. Stern (Theo- rem 2.2) that allowed us to avoid the previous use of Martin’s axiom in the proof of Theorem 2.6 and to give a ZFC proof of that.

Preliminaries and notation

We denote by Σ the setN^Nof infinite sequences of positive integers, endowed with the cartesian product topology, which makes Σ (usually called the “Baire space”) a Polish space (i.e., homeomorphic to a complete separable metric space).

We denote by S the set S∞

n=0Nⁿ (N⁰ = {∅}) of finite sequences of positive integers. We give the following partial order in S: for s = (s1, . . . , sn), t = (t1, . . . , tm) members of Swe define s≤tiffn≤mands_i=t_i fori= 1, . . . , n.

Ifs= (s₁, . . . , s_k)∈S,σ= (n₁, n₂, . . . , n_i, . . .)∈Σ andm∈Nthen we write, (i) s < σiffs_i=n_i fori= 1, . . . , kand

(ii) σ|mfor the finite sequence (n1, n2, . . . , nm).

For every s ∈ S, we set Vs = {σ ∈ Σ : s < σ}; it is easy to see that the countable family{V_s :s∈S} is a base for the topology of Σ consisting of open and closed sets.

For a (Hausdorff) topological spaceX,K(X) is the set of non empty compact subsets ofX. A mapF : Σ → K(X) is said to be upper semicontinuous iff for everyσ ∈ Σ and every open subset U of X with F(σ)⊆ U there existss ∈ S withs < σso thatF(V_s)⊆U.

LetXbe a (real) Banach space andX^∗its dual. ThenB_X, B_X∗will denote the closed unit balls ofX andX^∗ respectively. The Mackey topologyτ =τ(X^∗, X) is the finest locally convex topology onX^∗ whose dual space isX. The Mackey-

Arens theorem characterizesτ as the topology onX^∗ of uniform convergence on weakly compact (absolutely convex) subsets ofX (see [H-H-Z, pp. 163–165]).

1

Definition 1.1. Let X be a Banach space and K ⊆X. We shall say that K strongly generates X if for every weakly compact subsetLofX and everyε >0 there existsλ >0 such that

(1) L⊆λK+εB_X.

Remark 1.2. If a setK strongly generates a Banach spaceX then clearlyK is total in X. In the converse direction we observe that, if the convex symmetric set K is total in X then condition (1) of the previous definition holds for every norm-compact subset ofX. Indeed, if L⊆X is norm-compact and ε >0, pick x₁, . . . , x_n∈Lsuch thatL⊆Sn

i=1B(x_i, ε/2). Let now λ >0 andw₁, . . . , w_n∈ K with kx_i−λw_ik < ε/2 for i = 1,2, . . . , n, then it is easy to see that L ⊆ λK+εB_X.

A Banach spaceX is said to bestrongly weakly compactly generated (SWCG) if it is strongly generated by a weakly compact set K ([S-W]). We recall that a Banach space X is called weakly compactly generated (WCG), if it contains a weakly compact set K which is total in X since we may assume by Krein’s theorem thatKin addition is convex and symmetric, it follows from Remark 1.2 that every SWCG Banach space is WCG. The two notions are distinct. Indeed, every separable Banach space is WCG (let{x_n:n∈N}be a bounded total subset ofX, then the set{^x_nⁿ :n∈N} ∪ {0}is a norm-compact total subset of X). On the other hand, every SWCG Banach space is both WCG and weakly sequentially complete, so the space c₀ is not SWCG (see [S-W, Theorem 2.5]). It follows in particular that not every separable Banach space is SWCG.

Example 1.3(Examples of SWCG spaces [S-W]).

(i) Reflexive spaces (obvious).

(ii) Separable Schur spaces (letK be a norm compact convex symmetric set which is total inX, since by definition weakly compact subsets ofX are norm compact, the conclusion follows from Remark 1.2).

(iii) The spaceL¹(µ), for aσ-finite measureµ. Ifµis finite then it is proved, using Dunfford-Pettis’ criterion for weakly compact subsets ofL¹(µ), that the closed unit ball K of L^∞(µ) considered as a subset of L¹(µ), is a weakly compact set that strongly generates L¹(µ). It then follows from standard results that L¹(µ) is SWCG for anyσ-finite measureµ.

For further examples and results about SWCG property we refer the reader to [S-W].

We shall introduce now a new class of Banach spaces; we call it the class of strongly weaklyK-analytic (SWKA) Banach spaces, which is related to the class of SWCG as WCG spaces are related to weakly K-analytic (WKA) spaces ([T], [N], [M-N], [D-G-Z]). We recall that a Banach space X is said to be WKA if there exists an upper semicontinuous map F : Σ =N^N → K(X), whereK(X) is the set of weakly compact non empty subsets of X, so that F(Σ) =X (i.e., S

σ∈ΣF(σ) =X).

Definition 1.4. A Banach space will be called strongly weakly K-analytic (SWKA) if there exists an upper semicontinuous map F : Σ → K(X) so that (F(Σ) =X and with the further property), for every weakly compact subsetLof X there existsσ∈Σ withL⊆F(σ).

It clearly follows from the above that every SWKA space is WKA and also that every closed linear subspace of a SWKA space is SWKA itself. It is well known from a result of Preiss and Talagrand that every (subspace of a) WCG space is WKA ([T], [H-H-Z, Proposition 288]). The method of the proof of this result gives the similar result for the relation of SWCG and SWKA properties.

Proposition 1.5. Every closed linear subspace of a SWCG spaceX is SWKA.

Proof: It follows from the above remarks that it suffices to prove the assertion for a SWCG spaceX. LetK be a weakly compact (convex and symmetric) set that strongly generatesX. We consider the map F : Σ→K(X) defined by the rule

F(σ) =

\∞ n=1

σ(n)K+1 nB_X∗∗

,

whereσ∈Σ andB_X∗∗ is the closed unit ball ofX^∗∗endowed with weak* topol- ogy. It is easy to verify thatF is upper semicontinuous and that F(Σ) =X, in particularX is WKA (see the proof of the result of Preiss-Talagrand). Let now L⊆X be a weakly compact set; since K strongly generates X, there exists for everyn∈N,mn∈Nso thatL⊆mnK+_n¹B_X. Setσ= (m₁, m₂, . . . , mn, . . .),

then it is obvious thatL⊆F(σ).

Note. In fact the method of the proof gives the further property X =

\∞ n=1

[^∞

m=1

(mK+1

nB_X∗∗) ,

that is,X is aK_σδ subset of (X^∗∗, ω^∗) (see [Ta, Theorem 3.2]).

Remark 1.6. (1) H. Rosenthal has constructed a non WCG subspace X of the space (L¹[0,1]^c, µ), where µ the Lebesgue product measure ([R1]).

Therefore the SWCG property is not preserved by closed linear subspaces;

this space X is obviously SWKA from Proposition 1.5 above. We shall present later (in Section 3) a refinement of the construction of Rosenthal which yields a non SWCG subspace ofL¹[0,1]; this example answers in the negative a question posed in [S-W].

(2) If, for a Banach spaceX, there exists an upper semicontinuous map F : Σ→ K(B_X) such that (i)F(Σ) =B_X and (ii) for everyL⊆B_X weakly compact there isσ∈Σ withL⊆F(σ), thenXis SWKA. In order to make this clear, define a map Φ : Σ×N→ K(X) by the rule Φ(σ, n) =nF(σ).

It is easy to check that Φ makesX a SWKA space.

The following proposition gives a simple criterion to check the SWKA property;

its analogue for WKA has been proved by Talagrand ([T, Proposition 6.13]). In the statement of our proposition we need the following partial order of the Baire space Σ: Forσ, τ ∈Σ we letσ≤τ iffσ(n)≤τ(n) for alln∈N. We notice that every set of the form Σ(σ) ={τ ∈Σ :τ ≤σ} is compact in Σ and also that the family{Σ(σ) :σ∈Σ} dominates the compact sets in Σ.

Proposition 1.7. LetX be a Banach space. Then the following are equivalent.

(i) The spaceX is SWKA.

(ii) There exists a family (Wσ)_σ∈Σ of weakly compact subsets ofX (resp. of B_X)such that:

(a) X =S

σ∈ΣW_σ (resp.B_X =S

σ∈ΣW_σ);

(b) for everyσ, τ ∈Σ, ifσ≤τ thenW_σ⊆W_τ;

(c) for every L ⊆X (resp. L ⊆ B_X) weakly compact there is σ ∈ Σ withL⊆W_σ.

Proof: (i) ⇒(ii). Let F : Σ→ K(X) be an upper semicontinuous map which makesX SWKA. We set W_σ =F(Σ(σ)) for σ∈Σ (recall that Σ(σ) ={τ ∈Σ : τ ≤σ}). It is easy to see that the family (W_σ)_σ∈Σ satisfies assertion (ii) forX. It is obvious that the family (W_σ∩B_X) is the proper family forB_X.

(ii)⇒ (i). Let (W_σ)_σ∈Σ be a family of weakly subsets ofB_X that satisfies (a), (b) and (c) of assertion (ii). We set W_s = S

s<τW_τ^w, for s ∈ S and then F(σ) = T∞

n=1W_σ|n. It is then easy to prove that F satisfies the requirements of Remark 1.6(2) and hence X is SWKA (cf. also the proof of Proposition 6.13

in [T]).

At this point we state (for comparison with the above proposition but also because we shall use it) a nice characterization of Polish metric spaces due to Christensen ([Ch, Theorem 3.3, p. 58]).

Theorem 1.8(Christensen). LetM be a metric space. ThenM is Polish if and only if there exists a family {Mσ : σ ∈Σ} of compact subsets of M such that:

(a) if σ, τ ∈Σ, σ ≤τ thenMσ ⊆Mτ and(b) for every L ⊆M compact there existsσ∈ΣwithL⊆Mσ.

Now we recall the concept of a Polish Banach space. A Banach spaceX is said to be Polish if its closed unit ballB_X endowed with the weak topology is a Polish space. Since (B_X, w) is then metrizable it follows in particular that a Polish Banach space has separable dual. Obvious examples of Polish Banach spaces are separable reflexive spaces. The predualJT∗ of the James tree spaceJT is Polish (without being of course a reflexive space); the spacec₀ is not Polish although it has separable dual ([E-W], [R2]).

Proposition 1.9. Let X be a Banach space with separable dual. Then X is SWKA if and only if it is Polish.

Proof: Since X has separable dual its closed unit ball (B_X, w) is a separable metric space, therefore the result is an immediate consequence both of Proposi-

tion 1.7 and Theorem 1.8.

It follows in particular from this proposition and previous remarks, that the spacec0 is not SWKA. This observation can be generalized as follows:

Corollary 1.10. LetΩbe an infinite compact Hausdorff space. Then the Banach spaceC(Ω) is not SWKA.

Proof: Since Ω is infinite the space C(Ω) contains an isomorphic copy of c₀ which is not SWKA, but as we have noticed every subspace of a SWKA space is

again SWKA, thereforeC(Ω) is not SWKA.

Remark 1.10.1. As it is proved in [S-W, Theorem 2.5] every SWCG Banach space is weakly sequentially complete, hence every subspace of a SWCG has the same property. So we get, from Rosenthal’sℓ¹-theorem [R0], that a subspace of a SWCG Banach space not containingℓ¹ is reflexive. It follows in particular that a non reflexive Polish Banach space is not a subspace of a SWCG space.

The SWKA property for a Banach space is by definition a topological property of the weak topology of the space. There is no reason to restrict this concept only for the weak topology of Banach spaces; so we define the strongK-analyticity for every Hausdorff topological space.

Definition 1.11. A Hausdorff topological space X will be called strongly K- analytic, if there exists an upper semicontinuous mapF : Σ→ K(X) such that

(a) F(Σ) =X and

(b) for everyL⊆X compact there isσ∈Σ withL⊆F(σ).

Remark 1.11.1. (1) It is clear that every stronglyK-analytic topological space isK-analytic in the classical sense; we shall see later that the converse is not true (see [J-R] and [T] for more information aboutK-analytic topological spaces). It is also obvious that SWKA Banach spaces are those Banach spaces which are stronglyK-analytic in their weak topology according to the above definition.

(2) Every Polish metric space (M, d) is stronglyK-analytic; indeed, it is known that the spaceK(M) of the compact non empty subsets ofM with the Hausdorff metric is also Polish, (see [E, 4.5.22, p. 370]); so letf : Σ→ K(M) be a continu- ous onto map; then the mapF : Σ→ K(M) defined by the ruleF(σ) =f(Σ(σ)) has the desired properties. In the converse direction, we note that Theorem 1.8 easily implies that every stronglyK-analytic metric space is Polish (see also The- orem 1.12). It is easily verified that every ˇCech-complete and Lindel¨of topological space X (equivalently, X is homeomorphic to a closed subset of some product M ×Ω where M is a Polish metric space and Ω a compact Hausdorff space) is stronglyK-analytic.

The following result says in particular that the class of strongly K-analytic topological spaces is natural and not “artificial”. Before we state this result it is necessary to recall the definition of the Vietoris topology (otherwise, exponential or Hausdorff topology) on the space K(X) of compact non empty subsets of a Hausdorff topological spaceX. The Vietoris topologyτν onK(X) has as a basis consisting of subsets ofK(X) of the form:

β(V₁, . . . , Vn) =n

K∈K(X) :K⊆ [n i=1

V_i and K∩V_i6=∅ for i= 1, . . . , no where n ∈ N and V₁, . . . , Vn are open non empty subsets of X (see [E, 2.7.20, p. 162]).

In the proof of our result we shall make use of the following simple property ofK-analytic topological spaces: IfX is a Hausdorff and completely regularK- analytic topological space then there exists an upper semicontinuous map F : Σ→ K(X) such that

(i) F(Σ) =X and

(ii) ifσ, τ ∈Σ andσ≤τ thenF(σ)⊆F(τ) (see [T, pp. 409–410]).

Theorem 1.12. LetX be a Hausdorff and completely regular topological space.

Then the following are equivalent:

(i) X is stronglyK-analytic;

(ii) (K(X), τ_ν)is stronglyK-analytic;

(iii) (K(X), τ_ν)is K-analytic.

Proof: (i) ⇒ (ii) Let F : Σ → K(X) be an upper semicontinuous map with the further property that for every compact subsetK of X there isσ ∈Σ with K⊆F(σ). We define a set valued map Φ : Σ→ K(K(X)) by the rule

(1) Φ(σ) =K(F(σ)), σ∈Σ.

It then follows from standard properties of Vietoris topology that the set Φ(σ) is compact in (K(X), τν) for allσ ∈Σ and hence Φ is well defined (see [E], [K]

or [Ch]). Let Ω ⊆ (K(X), τ_ν) be compact. Set X_Ω = S

{K : K ∈ Ω}, then X_Ω (from the properties ofτ_ν) is a compact subset ofX and sinceX is strongly K-analytic (viaF) there isσ∈Σ such that X_Ω ⊆F(σ). It follows immediately that

(2) Ω⊆Φ(σ).

It is clear that it remains to show that Φ is an upper semicontinuous map. So let σ∈Σ andV be an open subset of (K(X), τ_ν) with

(3) Φ(σ)⊆V.

Since Φ(σ) is compact in (K(X), τν) we may assume that V is equal to a finite union of basic open sets. Assume for simplicity thatV is the union of two basic open sets, that is, V = V1∪V2 where V_k = β_k(V_1,k. . . , V_nk,k), k = 1,2 and V1,1, . . . , Vn1,1 and V1,2, . . . , Vn2,2 are open non empty subsets of X. It then follows from (3) and the definition of basic open sets of Vietoris topology that for everyK∈Φ(σ) we have that

(4) either

K∩V_i,16=∅ for i= 1,2, . . . , n₁ and K⊆

n1

[

i=1

V_i,1

(5) or

K∩V_i,26=∅ for i= 1,2, . . . , n₂ and K⊆

n2

[

i=1

V_i,2 .

We also have that for everyt∈F(σ)

(6)

either t∈

n1

[

i=1

V_i,1 or t∈

n2

[

i=1

V_i,2 .

It follows from (6) that ifK∈Φ(σ) then

(7) K⊆\ⁿ¹

i=1

V_i,1

∪\ⁿ²

i=1

V_i,2 .

Assume without loss of generality that (4) holds. Set

V0 =n

K∈ K(X) :K⊆h\ⁿ¹

i=1

V_i,1

∪\ⁿ²

i=1

V_i,2i

∩[ⁿ¹

i=1

V_i,1o .

ThenV₀ is a basic open set in (K(X), τ_ν) so that Φ(σ)⊆V₀ ⊆V₁∪V₂. Indeed, ifK∈Φ(σ) thenK⊆T_n1

i=1V_i,1 andK⊆ T_n1

i=1V_i,1

∪ T_n2

i=1V_i,2

from (4) and (7) respectively, hence K ∈ V0. Let now K ∈ V0; if K∩ Tn1

i=1V_i,1

6= ∅ then sinceK⊆S_n1

i=1V_i,1 we find that K∈V₁; if K∩ T_n1

i=1V_i,1

=∅ then obviously K⊆Tn2

i=1V_i,2 ⊆Sn2

i=1V_i,2 henceK∈V2.

SinceF is upper semicontinuous there is n0∈Nso that F(V_σ|n0)⊆h\ⁿ¹

i=1

V_i,1

∪\ⁿ²

i=1

V_i,2i

∩[ⁿ¹

i=1

V_i,1 ,

(whereV_σ|n0 ={τ ∈Σ :τ|n₀ =σ|n₀}). It follows easily that Φ(V_σ|n0)⊆V₀, hence Φ is upper semicontinuous.

(ii)⇒(iii). It is obvious.

(iii) ⇒ (ii). Let Φ : Σ → K(K(X)) be an upper semicontinuous map which makes (K(X), τν) a K-analytic topological space. Assume without loss of gene- rality that Φ has the further property

if σ, τ ∈Σ, σ≤τ then Φ(σ)⊆Φ(τ).

SetXσ =S

{K:K∈Φ(σ)}, forσ∈Σ, then it is clear that we have the following:

(a) X_σ is compact inX for allσ∈Σ, (b) ifσ, τ ∈Σ,σ≤τ thenX_σ⊆X_τ and

(c) if K ∈ K(X) then there is σ ∈ Σ withK ⊆X_σ (indeed, K ∈ Φ(σ) for someσ∈Σ, thusK⊆X_σ).

We shall show that the map F : Σ → K(X) : F(σ) = Xσ, σ ∈ Σ is upper semicontinuous. Letσ∈Σ andV be an open subset ofX withX_σ ⊆V, then the setβ(V) ={K∈ K(X) :K⊆V} is open in (K(X), τ_ν) and it is easily seen that Φ(σ)⊆β(V). Let nown₀ ∈Nbe such that Φ(V_σ|n0)⊆β(V); then it is easy to show thatF(V_σ|n0)⊆V.

The proof of the theorem is complete.

The following result gives some elementary stability properties of stronglyK- analytic topological spaces.

Proposition 1.13. (i)If X is any stronglyK-analytic topological space and Y is a closed subset of X thenY is also stronglyK-analytic.

(ii) If (X_n) is any sequence of strongly K-analytic topological spaces then their cartesian productX =Q∞

n=1X_n is stronglyK-analytic.

Proof: (i) Let F : Σ→ K(X) be an upper semicontinuous map such that for every compact subsetK ofX there existsσ∈Σ withK⊆F(σ). SetU =X\Y andV ={σ∈Σ :F(σ)⊆U}, then clearlyV is an open subset of Σ. We define

a map Φ : Σ\V → K(Y) by the rule Φ(σ) =F(σ)∩Y, σ ∈Σ\V; since the set Σ\V is closed in Σ it is a Polish space, hence there exists a continuous onto map f : Σ→Σ\Y. Now it is easy to verify that the map σ∈Σ→Φ(f(σ))∈ K(Y) makesY a stronglyK-analytic space.

(ii) Let F_i : Σ→ K(X_i),i ∈Nbe an upper semicontinuous map that makes X_i a stronglyK-analytic space. Set for every (σ₁, . . . , σ_n, . . .)∈Σ^N

F(σ₁, . . . , σ_n, . . .) = Y∞ i=1

F_i(σ_i).

The space Σ^Nis (obviously) homeomorphic to the space Σ and the mapF : Σ^N→ K(X) defined above compact valued. The mapF is also upper semicontinuous;

for a proof of this fact we refer the reader to [J-R, Theorem 2.5.4, p. 23]. Let K be a compact subset ofX; it is clear thatK ⊆Q∞

i=1K_i, where K_i =π_i(K), i ∈ N and π_i : X → X_i is the projection at coordinate i. Since K is compact K_i is compact for all i ∈ N; hence there is for every i ∈ N (using the strong K-analyticity ofX_i)σ_i ∈Σ such that K_i ⊆F_i(σ_i). It then clearly follows that K ⊆Q∞

i=1K_i ⊆F(σ1, . . . , σn, . . .), therefore F makesX a stronglyK-analytic

space.

Now we give some easy consequences of the above results concerning strong K-analyticity.

Corollary 1.14. Let X be a separable Banach space which is not SWKA (for instanceX =c0). Then the spaceK(X) (resp.K(B_X))of weakly compact subset ofX (resp. of B_X)with Vietoris topology(induced by its weak topology)is not K-analytic, in particular it is not analytic.

Proof: It is an immediate consequence of Theorem 1.12.

Note. We notice that every separable Banach space with the weak topology is an analytic space. Indeed, (X,k · k) is a Polish space and the identity map id : (X,k · k)→(X, w) is obviously continuous.

The following result says that a countable metric space is not necessarily strongly K-analytic. Since clearly a countable metric space is analytic, we get in particular that the class of stronglyK-analytic (metric) spaces is not stable under continuous images.

Proposition 1.15. The space of rational numbers Q(with the usual topology fromR)is not stronglyK-analytic.

Proof: Since (as it is well known) Q is not a Polish space, the result follows

immediately from Remark 1.11.1(2).

Remark 1.16. (1) It follows immediately from Proposition 1.13 that the class of SWKA Banach spaces is stable under closed subspaces and finite products. On the other hand it is not stable under continuous linear maps (c₀ is a quotient ofℓ¹).

(2) If a Banach space X has a ˇCech-complete unit ball in its weak topology (that is,B_X is aG_δ subset of (B_X∗∗, ω^∗)) thenX is SWKA. This is so, because then X is isomorphic to Y ⊕Z, where Y is a Polish andZ a reflexive Banach space (see [E-W] and [H-H-Z, Theorem 315]).

(3) It follows immediately from Theorem 1.12 that a Banach spaceX is SWKA iff the spaceK(X) with Vietoris topology (induced by the weak topology ofX) isK-analytic.

2

In this section we are concerned with separable SWKA Banach spaces which do not containℓ¹. We shall show that such Banach spaces have separable duals and thus they are Polish according to Proposition 1.9. We shall need for this purpose a nice generalization of Rosenthal’sℓ¹-theorem due to Stern ([S, Theorem 2.1], see also [To, Theorem 12]); we also need a deep result of Stegall, that Stegall used in his study of dual Banach spaces with Radon-Nikodym property (RNP) (see [St, Theorem 2.3]). It should be recalled that in the special case when the separable spaceX (which does not contain ℓ¹) is a subspace of a SWCG space thenX is reflexive (see Remark 1.10.1).

The following result from [M, Remark 1.4.1] will be used in the sequel; we prove it for completeness (the proof given in [M] is different).

Lemma 2.1. Let Γ be a non empty set and(Γσ)_σ∈Σ a family of subsets of Γ such that

(i) Γ =S

σ∈ΣΓσ,

(ii) if σ, τ ∈Σandσ≤τ thenΓσ ⊆Γτ and (iii) eachΓσ is a finite set.

ThenΓis at most countable.

Proof: We set Γ_∅ = Γ and Γs = S

s<σΓσ for s ∈ S. Assume that Γ is un- countable. Since Γ = Γ_∅ = S∞

n=1Γn there is n₁ ∈ N such that |Γn1| ≥ ω⁺; since Γn1 = S∞

n=1Γn1n there is n2 ∈ N with |Γ_n1n2| ≥ ω⁺. We proceed by induction and find a sequence n1, n2, . . . , n_k, . . . of natural numbers such that Γn1 ⊇ Γn1n2 ⊇ . . . ⊇ Γn1n2...nk ⊇ . . ., and |Γ_n1n2...nk| ≥ ω⁺ for k ∈ N. Set σ= (n1, n2, . . . , n_k, . . .)∈Σ, then the sequence of sets (Γ_σ|k)_k∈N is decreasing and consists of uncountable sets. Pick a sequence (γ_k) of distinct points of Γ and (σ_k)⊆Σ such thatσ|k < σ_kandγ_k∈Γσk fork∈N. It is clear that the sequence (σ_k) is convergent in the space Σ and σ_k →σ, thus the set {σ_k : k∈N} ∪ {σ}

is compact in Σ. Letτ ∈ Σ withσ_k ≤τ for allk ∈ N, therefore Γσk ⊆Γτ for

allk ∈N. It clearly follows that{γ_k: k∈N} ⊆Γ_τ; but this is a contradiction because the set{γ_k:k∈N}is infinite and the set Γ_τ is finite.

Throughout this paper T stands for the dyadic tree; i.e., T = S∞

n=0{0,1}ⁿ ordered (as the treeS =S_∞

n=0Nⁿ) by the relation “sis an initial segment oft”, denoted bys≤t. By asubtreeT^′ ofT we mean any subset ofT having a unique minimal element and such that any its element has exactly two immediate succes- sors; thus in particularT^′is isomorphic toT. By the term chain (resp. antichain) of T we mean a set of pairwise comparable (resp. incomparable) elements ofT; a branch of T is any maximal chain of T. Note that if T^′ is any subtree of T then the chains and antichains ofT^′ are chains and antichains ofT.

Theorem 2.2(Stern). LetX be a Banach space and(xs)_s∈T a bounded family of elements of X; then there exists a subtreeT^′of Tsuch that one of the following alternatives holds:

(i) for any branchδ^′ ofT^′, the sequence(x_δ′|n)is weak-Cauchy;

(ii) for any branch δ^′ of T^′ the sequence (x_δ′|n) is equivalent to the usual ℓ¹-basis.

Denote by ∆ the Cantor set, i.e., ∆ = {0,1}^N. A family (hs)s∈T ⊆ C(∆) is said to be aHaar systemon ∆ if the following hold:

(i) for every s ∈ T, h_s = X_As where A_s is a non empty open and closed subset of ∆;

(ii) A_∅ = ∆ and for everys∈T,As=A_s0∪A_s1 andA_s0∩A_s1=∅;

(iii) for every δ ∈ ∆ the intersection T_∞

n=1A_δ|n is one point set (see [St, p. 215]).

Theorem 2.3(Stegall). LetX be a separable Banach space with non separable dual. Then for anyε >0there exist a subset∆_ε of the unit sphere of X^∗ which is weak* homeomorphic to the Cantor set∆, a Haar system(h_s)_s∈T on∆ and a family(es)_s∈T ⊆X withkesk ≤1 +εfor alls∈T such that if Φ :X →C(∆ε) is the canonical evaluation operator (i.e., Φ(x)(x^∗) = x^∗(x), for x ∈ X and x^∗ ∈∆ε), then

(1)

X∞ n=0

X

s∈{0,1}ⁿ

kΦ(e_s)−hsk

< ε.

Note. In the following lemmas until to the proof of our main result (Theo- rem 2.6), X stands for a separable Banach space with non separable dual not containingℓ¹ and (e_s)_s∈T, (h_s)_s∈T, Φ, for the families and the operator of Ste- gall’s theorem.

Lemma 2.4. Let(t_n)be a chain and(s_n)an antichain of T. Then the sequence (e_sn−e_tn)of X has no weakly convergent subsequence.

Proof: Assume without loss of generality that the given sequence is itself weakly convergent inX, sayw−lim(esn−etn) =x, hencew−lim Φ(esn−etn) = Φ(x).

Then we have:

kΦ(e_sn−etn)−(hsn−htn)k=k(Φ(e_sn)−hsn)−(Φ(etn)−htn)k

≤ kΦ(esn)−hsnk+kΦ(etn)−htnk →0, for n→ ∞

(the fact that this limit is zero is a consequence of inequality (1) of Stegall’s theorem). It then follows thatw−lim(h_sn−h_tn) = Φ(x) which is a contradiction because clearly the sequence (h_sn) is weakly null and the sequence (h_tn) is weakly

Cauchy but not weakly convergent.

Lemma 2.5. Let T^′ be a subtree of T such that for every branch δ of T^′ the sequence(e_δ|n)is weakly Cauchy. Then the Banach spaceX is not SWKA.

Proof: Assume for the purpose of contradiction that X is SWKA and let (W_σ)_σ∈Σ be a family of weakly compact sets in X satisfying assertion (ii) of Proposition 1.7. It is clear that the set of branches ∆^′ ofT^′can be identified with the (Cantor) set{0,1}^N; we associate with every branchδof T^′ the weakly null sequence

ϕ(δ) ={e_δ|n−1−e_δ|n:n∈N}.

It is obvious that for everyδ∈∆^′ there existsσ_δ∈Σ such that ϕ(δ)⊆W_σδ.

Set forσ∈Σ,Xσ ={δ∈∆^′ :ϕ(δ)⊆Wσδ}; it follows then from the properties of the family (Wσ)_σ∈Σ that

(i) ∆^′=S

σ∈ΣXσ,

(ii) ifσ, τ ∈Σ andσ≤τ thenXσ⊆Xτ.

Since the set ∆^′ is uncountable, Lemma 2.1 implies that there isσ₀ ∈Σ so that the setXσ0 is infinite. Let (δ_k) be a sequence of distinct points ofXσ0 which is convergent in the Cantor set ∆^′, sayδ_k→δ. Set d(δ_k, δ) = _m¹_k, k∈N(whered is the usual metric in the Cantor set ∆^′ ={0,1}^N) and assume without loss of generality that 2≤m₁< m₂< . . . < m_k< . . ., fork∈N. It is clear that

δ|_mk−1=δ_k|_mk−1 and δ_k(m_k)6=δ(m_k), for k∈N.

So we get that the sequenceε_k=δ_k|m_k,k∈Nconsists of pairwise incomparable elements ofT^′(and thus ofT) and also that,eεk−e_δ|mk−1=e_δk|mk−e_δk|mk−1,

for every k ∈N. But this is a contradiction because on one hand the sequence e_δk|mk−e_δk|mk−1,k∈N, is contained in the weakly compact setW_σ0 and on the other hand the sequenceeεk−e_δ|mk−1,k∈N, cannot have a weakly convergent

subsequence by Lemma 2.4.

Theorem 2.6. Let X be a separable Banach space not containing ℓ¹. If X is SWKA thenX has separable dual and thus is a Polish Banach space.

Proof: Assume that X has non separable dual and let (e_s)_s∈T be the family given in Stegall’s theorem. We apply Stern’s theorem for (e_s)_s∈T; sinceX does not containℓ¹ the first alternative of this result must hold, but this contradicts our assumption according to Lemma 2.5. So the spaceX has separable dual and

thus is Polish by Proposition 1.9.

Corollary 2.7. Every SWKA Banach spaceX not containingℓ¹ is Asplund and (hence)WCG.

Proof: Every separable subspace ofX is Polish by Theorem 2.6 and thus it has separable dual, so X is Asplund. As X is Asplund and WKA it is WCG (see

Corollary 4.4, Chapter VI of [D-G-Z]).

By using Stern’s theorem and the techniques used before we can also prove the following

Theorem 2.8. LetX be a Banach space and(e_s)_s∈T be a bounded family of X. Assume that

(i) for no chain(tn)of T the sequence(etn)is weakly convergent and (ii) for every antichain(sn) of T there is a subsequence (s^′_n)of (sn)so that

the sequence(e_s′

n)is weakly convergent.

ThenX is not SWKA.

Proof: We first remark that as it follows from (i) and (ii), for every pair (t_n), (s_n) where (t_n) is a chain and (s_n) an antichain ofT the sequence (e_sn−e_tn) of X has no weakly convergent subsequence.

Let T^′ be a subtree of T given by Stern’s result so that one of the following alternatives holds.

(1) For every chain (t_n) ofT^′, (e_tn) is a weak Cauchy sequence inX.

(2) For every chain (tn) ofT^′, (etn) is equivalent to the usualℓ¹-basis.

Assume without loss of generality thatT^′=T.

If alternative (1) holds, then we proceed in exactly the same way as in the proof of Lemma 2.5 and (by using the above mentioned remark) show thatX is not SWKA.

Now assume that (2) holds forT. We notice that a dyadic tree contains un- countable many antichains. We give a proof of this (simple) argument which has been taken from [S-W, Example 2.6, p. 391]. Letθ(0) = 1 andθ(1) = 0; we define

for δ= (δ_n)∈ {0,1}^N a sequence ε₁(δ) = (δ₁), ε₂(δ) = (θ(δ₁), δ₂), . . . , ε_n(δ) = (θ(δ₁), . . . , θ(δ_n−1), δ_n), . . . in T. Then (ε_n(δ)) is (obviously) an antichain ofT and the mapδ∈∆→(ε_n(δ))∈T^Nis 1−1.

Now we assume for the purpose of contradiction thatX is SWKA and proceed in a similar way as in the proof of Lemma 2.5. So let (Wσ)_σ∈Σ be a family of weakly compact subsets of X that satisfies assertion (ii) of Proposition 1.7. It follows clearly from our hypothesis that for every antichain (s_n) of T the set {e_sn : n ∈N} is weakly relatively compact in X, hence it is contained in some W_σ. So it follows in particular that for every δ ∈ {0,1}^N there is σ_δ ∈ Σ such that {e_εn(δ) :n ∈N} ⊆ W_σδ. Set for σ∈Σ, X_σ ={δ∈ {0,1}^N:{e_εn(δ) : n∈ N} ⊆Wσ}; it is then clear that

(a) {0,1}^N=S

σ∈ΣXσ and

(b) ifσ, τ ∈Σ andσ≤τ thenXσ⊆Xτ.

So it follows from Lemma 2.1 that there isσ0 ∈Σ withXσ0 an infinite set. Let (δn) be a non trivial sequence in Xσ0 which is convergent in the Cantor space, sayδn→δ. It is now easy to choose inductively an infinite chain{ε_k:k∈N} ⊆ S∞

n=1{ε_m(δn) : m∈N}, thus{e_εk :k ∈N} ⊆Wσ0. But this is a contradiction because the setW_σ0 is weakly compact and the sequence{e_εk:k∈N}equivalent

to the usualℓ¹-basis.

As an application of Theorem 2.8 we show that a weakly sequentially complete, (even separable) space need not be SWKA; the exampleX₀ which we are going to consider is due to Batt and Hiermeyer. It is proved in [S-W] thatX₀ is not SWCG. We show here that X₀ is not even SWKA, in particular X₀ is not a subspace of a SWCG space.

Example 2.9. Letαbe the set of finite chains of the dyadic tree T; it is clear that ifA⊆B∈αthenA∈α.

Let

X₀={f:T →R,kfk<+∞}, where

kfk= supnh X

i∈I

(X

s∈Ai

|f(s)|)²i1/2

:I finite, A_i∩A_j =∅, i6=j

and A_i∈α for i∈Io (see [A-M, Definition 3.13]).

It is easy to verify that (X₀,k · k) is a Banach space, having the set{es:s∈T} as an unconditional boundedly complete (normalized) basis. Therefore X₀ is a weakly sequentially complete separable dual space (it has the Radon-Nikodym property) and it does not containc₀ (see [L-T, Theorem 1.c.10]).

LetA⊆T. Then it easily follows from the definition of the spaceX₀ that:

(i) ifAis a chain ofT then the family{e_s:s∈A} is equivalent to the usual ℓ¹-basis;

(ii) if A is an antichain of T then {e_s : s ∈ A} is equivalent to the usual ℓ²-basis, hence in particular it is a weakly relatively compact subset of X₀.

So we get immediately from (i), (ii) and Theorem 2.8 thatX₀cannot be a SWKA space.

3

Our aim in this last section is to give an exampleX of a subspace of L¹[0,1]

(more exactly of the space L¹[0,1]^N) which is not SWCG. It should be noticed that since X is a subspace of a SWCG space, it is itself a SWKA space (see Proposition 1.5). This example answers in the negative a question posed in [S-W, Question (A), p. 397]: “Must a WCG subspace of an SWCG space again be SWCG? Specifically, must every subspace of a separable SWCG space again be SWCG? This is of particular interest for the spaceL¹[0,1].”

Our construction of the space X is similar and it is based on the classical construction of Rosenthal of a subspaceY of L¹[0,1]^c which is not WCG. This example answered (in the negative) the “heredity” problem in the class of WCG spaces (see [R1]). It is clear that our example has the similar property for the class of separable SWCG spaces.

The following result is crucial for our purposes.

Proposition 3.1. Let X be a Banach space with a normalized unconditional basisΓ. If X is SWCG then there exists a countable family(Γn)n of subsets of Γsuch that:

(i) Γn is a weakly relatively compact subset of Γ for alln ∈N (hence each Γn∪ {0}is weakly compact);

(ii) if Ais any weakly relatively compact subset of Γthen there isn∈Nwith A⊆Γn.

Proof: LetKbe a weakly compact convex symmetric subset ofX that strongly generatesX (see Definition 1.1). Assume without loss of generality thatK⊆B_X. We consider the operator T : X^∗ → C(K) defined by the ruleT(x^∗) = x^∗|_K, x^∗ ∈X^∗. It is easy to see thatTis a bounded linear one-to-one operator (kTk ≤1) with the further property that its restriction onB_X∗ is a weak* to pointwise con- tinuous map. Therefore the compact space (B_X∗, w^∗) is affinely homeomorphic to the compact space (T(B_X∗), τp), where τp is the topology of pointwise con- vergence on C(K). Since the set T(B_X∗) is bounded in C(K) it follows from Grothendieck’s theorem (see [R1, p. 102]) that the topologies of pointwise conver- genceτp and the weak topology coincide onT(B_X∗); it follows in particular that

T is weakly compact (cf. the proof of Corollary 3.4 of [R1]). As it is proved in [S-W, Theorem 2.1], a Banach spaceX is SWCG if and only if the space (B_X∗, τ), whereτis the Mackey topology ofX^∗is metrizable. It follows from the method of the proof of implication (c)⇒(a) of Theorem 2.1 that the norm-metric ofC(K) metrizes the Mackey topology ofB_X∗, that is, (B_X∗, τ)≈(T(B_X∗),k · k₁), where k · k₁ denotes the supremum norm ofC(K).

We set for everyn∈N, Γ_n={γ ∈Γ : kT(γ^∗)k₁≥ ¹_n}, whereγ^∗ denotes the biorthogonal functional ofγ∈Γ.

Now, by using the fact that T is a weakly compact operator and following the method of the proof of implication (4) ⇒ (1) of a result of Johnson (see Remark 3.2 below) we can prove that every set Γn, n ∈ N, is weakly relatively compact inX and also that Γ =S∞

n=1Γn. We shall show that this is the desired family.

Claim. LetA ⊆Γ. Then A is a weakly relatively compact subset ofX if and only if there isδ >0 withkT(γ^∗)k₁ ≥δfor allγ∈A.

Proof of the Claim: Assume that A is a weakly relatively compact subset of X and also assume for the purpose of contradiction that there is a sequence γ₁, . . . , γn, . . . of distinct points ofAso that limn→∞kT(γ_n^∗)k₁ = 0, equivalently τ−limn→∞γ_n^∗ = 0. Since the Mackey topology ofX^∗coincides with the topology of uniform convergence on weakly compact subsets ofX we conclude that,

an= sup{|γ_n^∗(x)|:x∈A} →0, for n→ ∞,

becauseA is a weakly relatively compact subset ofX. On the other hand, since γ_n ∈ A for n ∈ N, we get that γ_n^∗(γ_n) = 1 for all n ∈N, hence a_n ≥ 1 for all n∈N, which is a contradiction.

For the “if” assertion of the Claim consider a positiveδso thatkT(γ^∗)k₁≥δ, for everyγ∈A. Pick n0∈Nwith _n¹₀ ≤δ, then it is obvious thatA⊆Γn0 and thusAis also a weakly relatively compact set.

It is clear that this Claim finishes the proof of the proposition.

Remark 3.2. The result of Johnson mentioned in the proof of Proposition 3.1 states that: For a Banach spaceX with an unconditional basis Γ,X is WCG if and only if there exists a sequence (Γ_n) of subsets of Γ such that, Γ = S∞

n=1Γ_n and each Γ_nis a weakly relatively compact subset ofX(see [R1, Proposition 1.3]).

This result was used for the proof of some properties of Rosenthal’s example.

Let (X,M, µ) be a probability measure space. Following Rosenthal [R1], we define for everyf ∈L¹(µ) the modulus of absolute continuityω(f, δ) off by the rule

ω(f, δ) = sup{

Z

E

|f|dµ:E∈ M, µ(E)≤δ}, δ∈(0,1].

So a functionω(f,·) : (0,1]→Ris defined which has the following properties:

(i) ω(f,·) is increasing on (0,1] and (ii) lim_δ→0ω(f, δ) = 0 (see [R1, p. 89]).

Using the functionω(f,·) the classical characterization of relatively weakly com- pact subsets ofL¹(µ) may be reformulated as

Lemma 3.3. LetS be a non empty bounded subset ofL¹(µ). Then S is rela- tively weakly compact if and only if lim_δ→0[sup_f∈Sω(f, δ)] = 0 (see Lemma1.4 of [R1]).

Now we let, R=n

r: [0,1]→R, r∈L¹[0,1], Z 1

0

r dx= 0 and Z 1

0

|r|dx= 1o (see [R1, p. 86]).

The following result is well known (see [R1, p. 90]). We give a (different) proof of this for completeness.

Lemma 3.4. The setRis not aσ-relatively weakly compact subset of L¹[0,1]

(that is,Ris not a countable union of relatively weakly compact sets inL¹[0,1]).

Proof: We define a map Λ : L¹[0,1] → R by the rule Λ(f) = R₁

0 f dx, f ∈ L¹[0,1]. It is clear that Λ is a non trivial continuous linear functional onL¹[0,1], therefore the setZ ={f ∈L¹[0,1] : Λ(f) = 0}is a closed hyperplane ofL¹[0,1].

The set R is obviously the unit sphere S_Z of Z. Since Z is not reflexive (Z is isomorphic toL¹[0,1]) the conclusion follows immediately from the following simple result: A Banach spaceX is reflexive if and only if its unit sphereS_X is

aσ-relatively weakly compact set.

Remark 3.5. It is clear that the set R endowed with the norm-metric is a complete separable metric space.

Letµdenote the product Lebesgue measure on the compact space Ω = [0,1]^R. We associate with each functionr∈ Rtheµ-integrable functionfr: [0,1]^R→R defined byf_r=r◦π_r, whereπ_r is the projection to therth coordinate (see [R1, p. 86]).

It is easy to verify that for everyr∈ Rwe have

(1) (i)

Z

Ω

frdµ= 0 and (ii)

Z

Ω

|fr|dµ= 1.

The following two lemmas are due to Rosenthal (see [R1, pp. 89–90] and the references given there).

Lemma 3.6. The family Re = {f_r : r ∈ R} is an unconditional basis for its closed linear spanY in the spaceL¹(µ) (with unconditional constant≤2).

Lemma 3.7. (i)If r∈ Rthenω(r, δ) =ω(fr, δ)for allδ∈(0,1], thus

(ii) if S ⊆ Rthen S is a relatively weakly compact subset ofL¹[0,1]if and only if Se={f_r:r∈S}is a relatively weakly compact subset of L¹(µ).

We notice that it is easy to prove assertion (i). Assertion (ii) is a consequence of (i) and Lemma 3.3.

Remark 3.8. It follows from Lemmas 3.4 and 3.7 that the set Re is not a σ- relatively weakly compact subset ofL¹(µ).

Let now ∆ be a norm-dense subset of the complete separable metric space Rendowed with the norm metric (see Remark 3.5). Set ∆ =e {f_r: r∈ ∆} and denote byY_∆the closed linear span of∆ ine L¹(µ); also denote byY the spaceY_R. Then the following result is proved.

Theorem 3.9. (i) (Rosenthal [R1]). The spaceY is not WCG.

(ii) The spaceY_∆is not SWCG.

Proof: We are interested in assertion (ii) of course but for completeness we also give the easy proof (due to Rosenthal) of assertion (i). The set Re is from Lemma 3.6 an unconditional basis ofY and, by Remark 3.8, it is not aσ-relatively weakly compact subset of L¹(µ). Now the result of Johnson (see Remark 3.2) proves immediately (i).

In order to prove (ii) assume for the purpose of contradiction that the space Y_∆ is SWCG. Since the family∆ is an unconditional basis ofe Y_∆ there exists by Proposition 3.1 a sequence (∆_n) of subsets of ∆ such that:

(a) each ∆e_n is a relatively weakly compact subset of Y_∆ (hence, ∆_n is a relatively weakly compact subset ofL¹[0,1]);

(b) ifK ⊆∆ andKe is a relatively weakly compact subset ofY_∆ then there isn₀∈Nso thatKe ⊆∆e_n0 (hence,K⊆∆_n0).

Denote by En the weak closure of ∆n in L¹[0,1] for n ∈ N, hence each En

is weakly compact. We shall show that R =S∞

n=1(R ∩En); since each of the setsR ∩En is a relatively weakly compact subset ofL¹[0,1], this equality clearly contradicts Lemma 3.4. So let r ∈ R; since ∆ is norm-dense in R there is a sequence (rn) ⊆ ∆ with limn→∞krn−rk₁ = 0. Therefore the set {rn : n ∈ N} ∪ {r}is norm and thus weakly compact subset ofL¹[0,1]. It follows then from Lemma 3.7(ii) that the set{f_rn:n∈N}is a relatively weakly compact subset of L¹(µ) and thus ofY_∆. So there exists (from (b))n₀∈Nwith{f_rn:n∈N} ⊆∆en0

which implies that{r_n : n ∈N} ⊆∆n0 and hence w−limrn =r ∈En0. The obvious inclusionS∞

n=1(R ∩En)⊆ R, finishes the proof of assertion (ii).

Corollary 3.10. Let∆be a countable norm dense subset of R. Then the space Y_∆ is a non SWCG subspace of the spaceL¹[0,1]^N(∼=L¹[0,1]).

Proof: It is an obvious consequence of the construction of Y_∆ and of Theo-

rem 3.9(ii).

Note. The spaceY_∆(where ∆ is a norm-dense subset ofR) is weakly sequentially complete as a subspace ofL¹[0,1]^∆ and it has an unconditional basis. Therefore its basis is boundedly complete andY_∆is isomorphic to a dual Banach space (see [L-T, Theorem 1.c.10]).

H. Rosenthal has proved, assuming Martin’s axiom plus the negation of contin- uum hypothesis (MA +⌉CH), the following result: If µis a probability measure on some measurable space andY is a closed linear subspace ofL¹(µ) of density character dimY <c, thenY is WCG (see Theorem 2.7 of [R1]). The first named author has proved several years ago (it was about 1993) a generalization of this result, which still remains unpublished. We decided to include this result in this note because it improves considerably the result of Rosenthal and also because its proof is short and sweet.

We shall need a definition and also the main result of [M] (see also [M-N]

and [D-G-Z]).

Definition 3.11 ([M, Definition 1.1]). For a topological spaceX we denote by c₁(X) the closed linear subspace ofℓ^∞(X) which consists of all bounded functions f : X → Rsuch that for every ε > 0 the set σ_ε(f) ={t∈ X : |f(t)| ≥ε} is a closed and discrete subset ofX.

Theorem 3.12 ([M, Theorem 4.1]). A Banach space Y is WKA if and only if there exists a bounded linear one-to-one operatorT :Y^∗→c₁(Σ×{0,1}^a), where dimY ≤2^a, which is weak* to pointwise continuous.

Note. It is obvious that the space Σ× {0,1}^ω is homeomorphic to the Baire space Σ, hence if dimY ≤2^ω =c, then the operator T of Theorem 3.12 takes values in the spacec1(Σ).

Now we state the above mentioned generalization of Rosenthal’s result.

Theorem 3.13 (MA +⌉CH). LetY be a WKA Banach space withdimY <c. ThenY is WCG.

Proof: SinceY is a WKA Banach space with dimY <cthere exists according to Theorem 3.12 a bounded linear one-to-one operatorT :Y^∗ →c₁(Σ) which is weak* to pointwise continuous. Therefore the space Ω =T(B_Y∗) endowed with the topology of pointwise convergence is homeomorphic to the compact space (B_Y∗, w^∗). Set Σ^′ =S{suppf :f ∈Ω}(where forf ∈c₁(Σ), suppf ={σ∈Σ : f(σ)6= 0}); then it is easy to see that the cardinality |Σ^′| of Σ^′ is equal to the topological weight of Ω and thus of (B_Y∗, w^∗); hence

|Σ^′|= dimY <c.

Now we recall a consequence of Martin’s axiom: Every subset Σ^′ of the Baire space Σ of cardinality smaller thanc is a σ-relatively compact subset of Σ (see [J-R, pp. 118–119]). So let (Kn) be a sequence of compact subsets of Σ with Σ^′ ⊆S_∞

n=1Kn. We set Γ =S_∞

n=1Kn and define a map Φ :Y^∗ →c₀(Γ) in the following way:

Φ(y^∗) =

T(y^∗)|_Kn n

n∈N

.

It is easy to verify that Φ is a well defined bounded linear one-to-one operator which is weak* to pointwise continuous. Since the range of Φ is the Banach space c₀(Γ) we conclude immediately from the classical theorem of Amir-Lindenstrauss thatY is a WCG Banach space (see [N], [M-N] and [D-G-Z]).

The following open questions concern a (non separable) SWKA Banach spaceX.

(a) Is X a subspace of a WCG Banach space, or at least a K_σδ subset of (X^∗∗, w^∗)? What happens if we assume furthermore that X is a dual Banach space?

(b) Assume thatX does not containℓ¹. Does thenX have a ˇCech-complete unit ball? (see Remark 1.16(2) and Corollary 2.7).

References

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[N] Negrepontis S.,Banach Spaces and Topology, Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, Amsterdam, 1984, pp. 1045–1142.

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